Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 263053, 8 pages
http://dx.doi.org/10.1155/2012/263053
Research Article

On the Dynamics of a Higher-Order Difference Equation

1Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 26 September 2011; Accepted 5 January 2012

Academic Editor: Taher S. Hassan

Copyright © 2012 H. El-Metwally. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the investigation of the following more general rational difference equation: 𝑦 𝑛 + 1 = 𝛼 𝑦 𝑛 βˆ‘ / ( 𝛽 + 𝛾 π‘˜ 𝑖 = 0 𝑦 𝑝 𝑛 βˆ’ ( 2 𝑖 + 1 ) ∏ π‘˜ 𝑖 = 0 𝑦 𝑛 βˆ’ ( 2 𝑖 + 1 ) ) , 𝑛 = 0 , 1 , 2 , … , where 𝛼 , 𝛽 , 𝛾 , 𝑝 ∈ ( 0 , ∞ ) with the initial conditions π‘₯ 0 , π‘₯ βˆ’ 1 , … , π‘₯ βˆ’ 2 π‘˜ , π‘₯ βˆ’ 2 π‘˜ βˆ’ 1 ∈ ( 0 , ∞ ) . We investigate the existence of the equilibrium points of the considered equation and then study their local and global stability. Also, some results related to the oscillation and the permanence of the considered equation have been presented.